Due to the importance of this formula for our work, we provide in Section  its detailed justification. In our approach, we take advantage of this explicit link with Euler and QGSW equations to prove the existence of periodic and quasi-periodic vortex patch motions for the Euler-\(\alpha \)  equations. In particular, we shall make appeal to the computations carried out in [1]}, [2]}, [3]}, [4]}, [5]}, [6]} relatively to these contexts. For the Euler-\(\alpha \)  model, the global existence and uniqueness of classical solutions has been obtained in [7]}. The well-posedness of weak solutions in the space of bounded Radon measures for (REF ) has been proved in [8]}. Indeed, the formula (REF ) and (REF ) imply that the vector field \(\mathbf {v}\)  is less singular than the Euler one, which allows to reach the class of measures for the well-posedness. An other consequence is that the Yudovitch theory also applies in this context and the weak solutions are Lagrangian. This fact is the starting point to be able to look for vortex patches. Before presenting the notion of vortex patch solutions together with the main results of this study, we may end this presentation by briefly mentioning that the 2D Euler-\(\alpha \)  model has also been intensively studied in the case of subset of \(\mathbb {R}^{2}\)  domains with suitable boundary conditions. We may for instance refer the reader to [9]}, [10]}, [11]}, [12]}, [13]}.
